Early on the morning of April 24, San Diego State head baseball coach Tony Gwynn and his staff gathered in his office to watch a baseball game. Not a typical time for baseball on the West Coast, but the early call was an easy one to answer as they watched one of their highest draft picks make his major league debut. It was just the first of many times this season the Hall of Famer would watch former Aztecs starter Justin Masterson pitch for the Red Sox.

“He’s a lot more mature, a lot more composed,” Gwynn said. “To me he’s learned how to pitch. At the college level, like most guys his size [6-foot-6], he could just overpower hitters. And with his stuff, he was able to do that for the most part. But now he’s learned to pitch and it looks like he’s trusting his stuff, and it looks like he’s having fun. It looks like he’s really enjoying himself and really trying to understand how to get guys out, and it looks like he’s really paying attention and doing a good job.

“When he was here, I think you could see that he had that kind of stuff where he would be successful at the big-league level, and I think he’s just scratching the surface.”

Masterson, 23, the Sox’ fifth pick in the 2006 draft, 71st overall, was called up from Double-A Portland that day for an emergency start as a bug worked its chaos in the clubhouse. Although he did not factor into the decision in that game, he showed a glimpse of the poise and stuff he would demonstrate this season, throwing six innings against the Angels at Fenway, giving up one run on two hits and four walks, striking out four. Masterson reeled off three wins before taking his first loss, putting together a record of 4-3 with a 3.67 ERA before moving into the bullpen after his last start July 5 in New York. In 27 appearances out of the pen, he posted a 2.36 ERA. In the postseason, he pitched 9„ innings, allowing two earned runs on 10 hits and five walks, with nine strikeouts, for a 1.86 ERA.

Does Gwynn think Masterson is best suited for the rotation or the bullpen?

“To me it’s too early, especially the way they used him late,” Gwynn said. “Late he was setting up for [closer Jonathan] Papelbon. Coming in in the eighth with his stuff, you could totally see that. And if you want to turn him into a starter now, you’re not sure if the stuff translates because in the eighth inning, he didn’t throw his changeup nearly as much as he did when he started a couple of games early. And when he was starting the one thing that seemed to happen quite a bit was his pitch count would get up, and he’d only be in the fifth, sixth inning.

“So the Red Sox are kind of the perfect club because they have enough of everything, where they can kind of take the time to figure out where exactly he best fits. But I think they found out this year that he fits. Somewhere in those 11, 12 guys he’s a guy that definitely will fit into whatever role they want to put him in.”

Masterson will likely hear his name churned through the trade rumor mill this offseason, as he did last year, when he was talked about as part of a package of prospects that would land left-hander Johan Santana in a Sox uniform.

“Business, business, business,” Gwynn said. “You don’t have any control right now. For him, I’m sure it’s about trying to get better. Whether he’s there in Boston or somewhere else, his game plan is going to be the same. You want to pitch in the big leagues. You want to be successful in the big leagues, and you want to do it for a long time. And I’m sure those are the questions he’s pondering.”

Masterson surprised Gwynn and his team last fall with an unannounced visit — and Gwynn would not be surprised if it happened again this year.

“He’s the type of guy that doesn’t want to shake your hand, he wants to give you a hug,” Gwynn said. “He’s absolutely the same guy, and you can’t help but love those type of guys because he really does it the right way. He does it with a great attitude. He does it with a smile on his face. And you know what? He’s competitive as heck, but you’d be hard pressed to know that if you met him off the field.”

Maureen Mullen covers the Red Sox for OT and can be reached at mmullen@globe.com